Wave Digital Filter Models
Perhaps the best known physics-based approach to digital filter design is wave digital filters, developed principally by Alfred Fettweis .A.9Wave digital filters may be constructed by applying the bilinear transform  to the scattering-theoretic formulation of lumped RLC networks introduced in circuit theory by Vitold Belevitch . Fettweis in fact worked with Belevitch.A.10Scattering theory had been in use for many years prior in quantum mechanics.
A key, driving property of wave digital filters is low sensitivity to coefficient round-off error. This follows from the correspondence to passive circuit networks. Wave digital filters also have the nice property of preserving order of the original (analog) system. For example, a ``wave digital spring'' is simply a unit delay, and a ``wave digital mass'' is a unit delay with a sign flip. The only approximation aspect is the frequency-warping caused by the bilinear transform. It is interesting to note that when it is possible to frequency-warp input/output signals exactly, a wave digital filter can implement a continuous-time LTI system exactly! See  for a discussion of wave digital filters and their relation to finite differences et al.
Transfer Function Models